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1

ASPECTS OF COULOMB GASES

DJALIL CHAFAÏ

Abstract. Coulomb gases are special probability distributions, related to potential theory, that appearat many places in pure and applied mathematics and physics. In these short expository notes, we focus onsome models, ideas, and structures. We present briefly selected mathematical aspects, mostly related toexact solvability and first and second order global asymptotics. A particular attention is devoted to two-dimensional exactly solvable models of random matrix theory such as the Ginibre model. Thematically,these notes lie between probability theory, mathematical analysis, and statistical physics, and aim to bevery accessible. They form a contribution to a volume of the Panoramas et Synthèses series around theworkshop États de la recherche en mécanique statistique, organized by Société Mathématique de France,held at Institut Henri Poincaré, Paris, in the fall of 2018 (https://statmech2018.sciencesconf.org/).

Contents

Notation 21. Coulomb electrostatics and equilibrium measures 22. Coulomb gases 62.1. One-dimensional log-gases as Coulomb gases 62.2. Beta-Ginibre gas 72.3. From multivariate statistics to atomic physics 72.4. The Wigner jellium and electrons in metals 72.5. Random polynomials 83. First order global asymptotics and large deviations 83.1. The large deviations principle 83.2. First order global asymptotics 93.3. Weakly confining versus strongly confining potential 93.4. Concentration of measure 94. Edge behavior 105. Global fluctuations and Gaussian free field 106. Aspects of general exact solvability 117. Exactly solvable two-dimensional gases from random matrix theory 137.1. Ginibre model 137.2. More determinantal models 208. Comments and open problems 21Acknowledgments 22References 22

There are several introductory texts around Coulomb gases. We refer for instance to [ER05, DG09,Dei99, AGZ10, ABDF11, For10] for the relation to random matrices, to [Ser15] for the relation to analysisand Ginzburg – Landau vortices, to [Bou15, GZ19b] and references therein for a relation to geometry, to[But17] and references therein for a relation to random polynomials, to [Rou15] for a relation to Fock –Hartree quantum theory and Bose – Einstein condensates, to [Ser18b] and [Lew21] for an overview froma mathematical analysis/physics perspective, and to [LACTMS19] for a statistical physics point of view.

Date: Summer 2021, compiled August 26, 2021.2000 Mathematics Subject Classification. Primary: 82B21; Secondary: 82D05; 81V45; 60F05.Key words and phrases. Asymptotic analysis; Exactly solvable model; High dimensional phenomenon; Mean-field particle

system; Random matrix; Spectral analysis; Determinantal point process; Coulomb gas; Potential theory; Wigner jellium;Langevin dynamics; Large deviations principle; Boltzmann – Gibbs measure.

1

2 DJALIL CHAFAÏ

Notation

The Euclidean norm of x = (x1, . . . , xd) ∈ Rd is |x| =√

x21 + · · · + x2

d. It is the modulus if d = 2 withthe identification C = R2. We set i = (0, 1) ∈ C. The real and imaginary parts of z ∈ C are denotedℜz and ℑz. The Lebesgue measure is denoted dx. Let (E, τ) be a topological space with Borel σ-fieldB(E). We denote by Cb(E,R) the set of bounded continuous functions E → R, and by M1(E) the set ofprobability measures on (E, B(E)). If µ1, µ2, . . . , µ are in M1(Rd), then limn→∞ µn = µ weakly, denoted

µnCb−→

n→∞µ, when for all f ∈ Cb(E,R) we have lim

n→∞

∫fdµn =

∫fdµ.

This defines a sequential topology on M1(E), giving a Borel σ-field B(M1(E)). A random probabilitymeasure on E is a random variable taking values in M1(E). By X ∼ µ we mean that the random variable

X has law µ. We denote byd= and d−→ the equality and the convergence in law respectively.

1. Coulomb electrostatics and equilibrium measures

The Coulomb kernel is identical to the Newton kernel. Mathematically, potential theory deals with theanalysis of the Laplacian and its Green function and the behavior of harmonic functions. In some sense, itemerges naturally from the gravitation theory of Johannes Kepler and Isaac Newton, as well as from themodeling of electrostatics, namely the study of the distribution of static electric charges on conductorsand their interactions. From this last point of view, it takes its historical roots in the works of Charles-Augustin de Coulomb, Joseph-Louis Lagrange, and Carl Friedrich Gauss. Some mathematical parts ofpotential theory were developed later on by – among others – Johann Peter Gustav Lejeune Dirichlet,Victor Gustave Robin, Henri Poincaré, David Hilbert, Charles-Jean de La Vallée Poussin [de 37], MarcelBrelot [Bre67], Otto Frostman [Fro35], Oliver Dimon Kellogg [Kel67], Gustave Choquet [Cho54], andtheir followers. Deep links with Markov processes and probability theory were explored in particularby Joseph Leo Doob in [Doo01], Gilbert Hunt, Claude Delacherie and Paul-André Meyer [DM78], andtheir followers. Nowadays the basic objects and structures of potential theory appear at many places inmathematics and physics, providing in general a very useful electrostatic modeling or interpretation.

Cet aspect probabiliste, que Brelot regrettait tant d’être arrivé trop tard pour maîtriser, est sans doutele plus bel exemple d’interaction féconde entre deux théories : La théorie du potentiel, née dans le ciel(Kepler 1618, Newton 1665) et la théorie des probabilités née d’un coup de dés (Pascal 1654), doncpresque simultanément, devaient après trois siècles et des petits pas l’une vers l’autre (Wiener 1923, puisP. Levy, Doob), prendre avec G. Hunt (1957) pleinement conscience que leurs parties les plus vivaces nesont que deux faces complémentaires d’un même bel objet, et qu’on ne peut bien comprendre l’une sansconnaître l’autre (le traité Dellacherie-P.A. Meyer veut en donner la preuve).

Gustave Choquet, La vie et l’oeuvre de Marcel Brelot (1903 – 1987) [Cho90].

Let d ≥ 1. The Coulomb kernel g in Rd is given by g(0) = +∞, and, for all x ∈ Rd \ {0},

g(x) =

log1

|x| if d = 2,

1(d − 2)|x|d−2

if not.

(1.1)

We say that (x, y) 7→ G(x, y) = g(x − y) is the Green function of the Laplace operator ∆ = ∂21 + · · · + ∂2

d ,and g is the fundamental solution of the Poisson equation, see for instance [LL01, Theorem 6.20]. Indeed,denoting δ0 the Dirac mass at the origin, we have, in the sense of Schwartz distributions,

− ∆g = cdδ0 and cd = dωd =2πd/2

Γ(d/2)(1.2)

where ωd is the volume of the unit ball (its surface is dωd). The case d = 3 is physical for electrostaticsmodeling in the ambient space. The case d = 2 also appears at many places in mathematical physics.The case d = 1 serves historically as a toy model, less singular but exactly solvable.

For simplicity, we suppose from now on that d ≥ 2.

ASPECTS OF COULOMB GASES 3

· · · · · ·1660 Newton, . . .1770 Coulomb, . . .1800 Gauss, . . .1870 Boltzmann, Gibbs, . . .1900 Thomson, de la Vallée Poussin, . . .1920 Fock, Hartree, . . .1930 Wigner, Wishart, . . .1940 Doob, Onsager, . . .1950 Choquet, Hunt, Wigner, . . .1960 Dyson, Ginibre, Mehta, Selberg, . . .1970 Kosterlitz, Landkof, Pastur, Thouless, . . .1980 Deift, Laughlin, Lebowitz, Saff, Voiculescu, . . .1990 Ben Arous, Edelman, Guionnet, Johansson, . . .2000 Forrester, Erdős, Lewin, Serfaty, Yau, . . .

· · · · · ·Table 1.1. The arrow of time and some of the main actors mentioned in the text orin the references. As mentioned in [Cha15], the Stigler law of eponymy states that “Noscientific discovery is named after its original discoverer.”, attributed by Stephen Stiglerto Robert K. Merton. This is also known as the Arnold principle by some people.

|x|

g(x)

10

1

−1

d = 3d = 2d = 1

Figure 1.1. Coulomb kernel in dimension 1 (solid line) 2 (dotted line) and 3 (dashed line).

For all µ ∈ M1(Rd) such that log(1 + |·|)1d=2 ∈ L1(µ), the Coulomb energy of µ is

E(µ) =12

∫∫g(x − y)dµ(x)dµ(y) ∈ (−∞, +∞]. (1.3)

Note that if d = 2 then E(µ) = +∞ if µ has a Dirac mass. If µ models the distribution of unit charges(say electrons) in Rd then E(µ) is the electrostatic self-interaction energy of the configuration µ.

We say that a Borel set B ∈ B(Rd) is of positive capacity when supp(µ) ⊂ B and E(µ) < ∞ for someµ ∈ M1(Rd), and is of zero capacity when it does not carry a probability measure µ with E(µ) < ∞.

For all µ ∈ M1(Rd) with log(1 + |·|)1d=2 ∈ L1(µ), the Coulomb potential of µ at x ∈ Rd is defined by

Uµ(x) =∫

g(x − y)dµ(y) = (g ∗ µ)(x).

4 DJALIL CHAFAÏ

We have Uµ(x) ∈ (−∞, +∞], and Uµ(x) = +∞ if µ has a Dirac mass at point x. We also have

E(µ) =12

∫Uµ(x)dµ(x). (1.4)

Since g ∈ L1loc(R2, dx), the Fubini – Tonelli theorem gives Uµ ∈ L1

loc(R2, dx), hence Uµ < +∞ almost

everywhere. Moreover Uµ = g ∗ µ, and, in the sense of Schwartz distributions, we get from (1.2) that

∆Uµ = −cdµ. (1.5)

In particular, this gives the formula

E(µ) =12

∫Uµdµ = − 1

2cd

∫Uµ∆Uµdx. (1.6)

When d ≥ 3, the functional E does not take negative values on probability measures because g ≥ 0.However, when d = 2, the functional E may take negative values on compactly supported probabilitymeasures, due to the change of sign of g when d = 2 inside and outside the unit ball. For instance if µr

is the uniform law on the circle {x ∈ C : |x| = r} of radius r > 0 then, for all x ∈ C2,

Uµr (x) = − log(r)1|x|≤r − log |x|1|x|>r, and E(µr) = − log(r)2

, (1.7)

which is negative if r > 1. See for instance [ST97, (0.5.5) and (I.1.6)] for these computations. Similary,if µR is the uniform law on the disc {x ∈ C : |x| ≤ R} of radius R > 0 then we find that for all x ∈ C2,

UµR (x) = −12

( |x|2R2

− 1 + 2 log R)

1|x|≤R − log |x|1|x|>R, and EµR =14

− log(R). (1.8)

The functionals U and E extend to signed measures. If η = µ − ν where µ and ν are two compactlysupported probability measures on R

d, then Uη vanishes at infinity and an integration by parts gives

E(η) =12

∫Uηdη = − 1

2cd

∫Uη∆Uηdx =

12cd

∫|∇Uη|2dx, (1.9)

see [Ser15]. This shows that E does not take negative values on signed measures with total mass zero.The right hand side of (1.9) is the “carré du champ” in potential theory [Rot76, Hir78] while −∇Uµ

is the electric field – “champ électrique” in French – generated by the configuration of charges µ.Let us introduce now V : Rd → (−∞, +∞] such that (we say then that V is an admissible potential):

• the function V is lower semi-continuous;• the set {x ∈ Rd : V (x) < +∞} has positive capacity;• the function V is not beaten by the Coulomb kernel at infinity, namely

lim|x|→∞

(V (x) − log |x| 1d=2) > −∞. (1.10)

The electrostatic energy with external potential V is defined from M1(Rd) to (−∞, +∞] by

EV (µ) =12

∫∫(g(x − y) + V (x) + V (y)) µ(dx)µ(dy). (1.11)

This makes sense since the function under the double integral is bounded below on Rd × Rd thanks to(1.10). Finally, for all µ ∈ M1(Rd), if both log(1 + |·|)1d=2 and V are in L1(µ), then

EV (µ) = E(µ) +∫

V (x)dµ(x). (1.12)

The external potential plays typically the role of a confinement.The convexity of the quadratic form EV is related to a Bochner positivity of the kernel g, see [Lan72,

HP00, CGZ14, BHS19]. Indeed for all λ ∈ (0, 1) and µ, ν ∈ M1(Rd) with E(µ) < +∞ and E(ν) < +∞and V ∈ L1(µ) ∩ L1(ν),

λEV (µ) + (1 − λ)EV (ν) − EV (λµ + (1 − λ)ν)λ(1 − λ)

= E(µ − ν) =1

2cd

∫|∇Uµ−ν |2dx ≥ 0.

We are now ready for the general concept of equilibrium measure and its properties. The followingcouple of theorems is a classic in potential theory. For a proof, we refer for instance to the books[Lan72, Hel14, ST97, Ser15] and to the articles [BAG97, CGZ14, Ser18b].

Theorem 1.1 (Equilibrium measure). The following properties hold true:

ASPECTS OF COULOMB GASES 5

Dimension d Potential V Equilibrium measure µV

≥ 1 ∞1|·|>r Uniform on sphere {x ∈ Rd : |x| = r}

≥ 1 finite and C2 With density ∆Vcd

on the interior of its support

(absolutely continuous part of µ∗)

≥ 1 12 |·|2 Uniform on unit ball with density 1|·|≤1

ωd

(Ginibre) 2 12 |·|2 Uniform on unit disc with density 1|·|≤1

π

(Spherical) 2 12 log(1 + |·|2) Heavy-tailed with density 1

π(1+|·|2)2

(CUE) 2 ∞1([a,b]×{0})c Arcsine on [a, b] × {0}, density s 7→ 1s∈[a,b]

π√

(s−a)(b−s)

(GUE) 2 |·|2

2 1R×{0} + ∞1(R×{0})c Semicircle on [−2, 2] × {0}, density s 7→√

4−s2

2π 1s∈[−2,2]

Table 1.2. Basic examples of equilibrium measures. Some other examples are given inSection 7.2. The last four examples appear as limiting spectral distributions of randommatrices. The last two examples are singular in the sense that the potential is infiniteoutside the real line. They appear as one-dimensional log-gases from random matrices.In this case, the equilibrium measure cannot be deduced as a specialization of the secondexample, and its computation is a bit more subtle, see for instance [ST97].

(1) EV is strictly convex on its domain, is lower semi-continuous, with compact level sets;(2) infM1(Rd) EV < +∞;

(3) there exists a unique µV ∈ M1(Rd), called the equilibrium measure, such that

EV (µV ) = infµ∈M1(Rd)

EV (µ) in other words µV = arg minM1(Rd)

EV .

Some examples of equilibrium measures are gathered in Table 1.2.

Theorem 1.2 (Properties of the equilibrium measure). The following properties hold true:

(1) the equilibrium measure µ is compactly supported if

lim|x|→∞

(V (x) − log |x| 1d=2) = +∞; (1.13)

(2) the equilibrium measure µV has finite Coulomb energy E(µV ) ∈ R;(3) we have supp(µV ) ⊂ {x ∈ Rd : V (x) ≤ R} for some constant R < ∞;(4) the following Euler – Lagrange equations hold:

• UµV (x) + V (x) ≤ cV for all x ∈ supp(µV ),• UµV (x) + V (x) ≥ cV for all x ∈ Rd except on a set of zero capacity,

where cV is a quantity called the modified Robin constant defined by

cV = E(µV ) −∫

V dµV .

In particular, for all x ∈ supp(µV ) except on a set of zero capacity, we have

UµV (x) + V (x) = c.

In particular, we have the equality in the sense of distributions

µV =∆V

cd,

and the interior of supp(µV ) does not intersect {∆V < 0}.

Remark 1.3 (Logarithmic kernels and Riesz kernels).

6 DJALIL CHAFAÏ

• The logarithmic kernel in dimension d is given by

− log |x| , x ∈ Rd, x 6= 0;

• The Riesz kernel ks in Rd with parameter s > 0 is given by

1s|x|s , x ∈ R

d, x 6= 0.

• The Coulomb kernel in dimension d 6= 2 matches the Riesz kernel with s = d − 2;• The logarithmic kernel for all d ≥ 1 can be seen as the Riesz kernel with s = 0. Indeed, it suffices

to remove the singularity in the sense that for all x ∈ Rd with x 6= 0,

lims→0

|x|−s − 1s

= ∂s=0|x|−s = − log |x|.

In particular the Coulomb kernel in dimension d = 2 is the Riesz kernel with s → 0.• For all α ∈ (0, d), the Riesz kernel with s = d − α is the fundamental solution of the fractional

Laplace operator ∆α = ∆α2 , a Fourier multiplier, non-local operator if α 6= 2, see [CGZ14, RS16].

We refer for instance to [BHS19] for more analytic properties of these kernels and various applications.

2. Coulomb gases

Let d ≥ 2, n ≥ 1, β > 0, and let g and V as before. Suppose moreover that V is such that∫

Rd

e−nβ(V (x)−log(1+|x|)1d=2)dx < ∞. (2.1)

By using the fact that g ≥ 0 when d ≥ 3 and |x − y| ≤ (1 + |x|)(1 + |y|) when d = 2, we get then

Zn =∫

(Rd)n

e−βEn(x1,...,xn)dx1 · · · dxn < ∞

where

En(x1, . . . , xn) = n

n∑

i=1

V (xi) +12

∑

i6=j

g(xi − xj).

The Coulomb gas Pn is the Boltzmann – Gibbs probability measure on (Rd)n given by

dPn(x1, . . . , xn) =e−βEn(x1,...,xn)

Zndx1 · · · dxn. (2.2)

It models a “gas of electrons” in Rd of charge 1/n, at positions x1, . . . , xn, inverse temperature βn2, energy(1/n2)En(x1, . . . , xn), subject to Coulomb pair interaction and external field of potential V , namely

βEn(x1, . . . , xn) = βn2( 1

n

n∑

i=1

V (xi) +1n2

∑

i<j

g(xi − xj))

. (2.3)

Beware that we should interpret Pn as a way to model a random static configuration of charged particles.We deal here with electrostatics rather than with electrodynamics. The charged particles do not moveand there is no magnetic field. We have only an electric field.

In view of Remark 1.3, we could also define log-gases and Riesz gases. We do not follow this idea inthese notes, for simplicity and because the Coulomb case is by far the most important in applications.

2.1. One-dimensional log-gases as Coulomb gases. The formula (2.2) makes sense provided thatZn > 0. Actually the integration in (2.1) should be interpreted as with respect to the trace of theLebesgue measure or Hausdorff measure on {V < +∞} ⊂ Rd. Similarly the integration in (2.2) should beinterpreted as with respect to the trace of the Lebesgue measure or Hausdorff measure on {V < +∞}n ⊂(Rd)n. This allows to incorporate in the Coulomb gas model (2.2) the one-dimensional log-gases of randommatrix theory, by taking d = 2 and V = +∞ on Sc where S is a one-dimensional subset of R2, typicallyS = {x ∈ R2 : x2 = 0} or S = {x ∈ R2 : |x| = 1}. This includes all beta Hermite/Laguerre/Jacobi

ASPECTS OF COULOMB GASES 7

ensembles, Gaussian Unitary/Orthogonal/Simplectic Ensembles, etc. For instance, the famous GaussianUnitary Ensemble (GUE) corresponds to take d = 2 and

x ∈ R2 7→ V (x) =

|x|22

if x ∈ S = R × {0}+∞ if not

.

For simplicity, we do not study further the one-dimensional log-gases, in particular the ones comingfrom random matrix theory. We refer to the books [Dei99, Meh04, ER05, DG09, For10, AGZ10, PS11].Actually, most of the models that we consider in the sequel are fully dimensional in the sense that V isfinite everywhere. The simplest models that we focus on are two-dimensional: beta-Ginibre gases.

2.2. Beta-Ginibre gas. The case d = 2 is known as the two-dimensional one-component plasma. Wecall it the beta gas for short. Its density with respect to the Lebesgue measure on (R2)n = Cn = R2n is

(z1, . . . , zn) ∈ Cn 7→ e−nβ

∑n

j=1V (zj)

Zn

∏

i<j

|zi − zj |β . (2.4)

The quadratic potential case V = 12 |·|2 is sometimes referred to as the beta-Ginibre gas. In the special

case β = 2 and V = 12 |·|2, that we call the Ginibre gas, the density of Pn can be written as

(z1, . . . , zn) ∈ Cn 7→ nnϕn(

√nz1, . . . ,

√nzn) with ϕn(z1, . . . , zn) =

e−∑

n

j=1|zj |2

πn∏n

k=1 k!

∏

i<j

|zi − zj|2. (2.5)

The beta gas (2.4) with V = 12 |·|2 and β ∈ {2, 4, 6, . . .} matches the squared modulus of the Laughlin

wave function of the fractional quantum hall effect [Lau87, Gir05]. The Ginibre gas (2.5) matches thedensity of the eigenvalues of Gaussian random matrices [Gin65] (see Section 7.1 for more details), thedistribution of vortices in the Ginzburg – Landau modeling of superconductivity [Ser15], and rotatingtrapped fermions in two dimensions [LACTMS19]. The beta gas (2.4) with β = 2 such as the Ginibre gas(2.5) has a determinantal structure which provides exact solvability (see Section 7.1 for more details).

2.3. From multivariate statistics to atomic physics. Historically, Coulomb gases emerged in math-ematical statistics in the years 1920/30 in the study of the spectral decomposition of empirical covariancematrices of Gaussian samples. We speak nowadays about Laguerre ensembles and Wishart random ma-trices. In the 1950s, Eugene P. Wigner discovered by accident this model when reading a statisticstextbook, and this led him to use random matrices for the modeling of energy levels of heavy nuclei inatomic physics, see for instance [Cha14, BW11]. His work generated an enormous trend of activity instatistical physics in the 1960s, with the works of Gaudin, Mehta, Dyson, Ginibre, Marchenko, Pastur,among others. The term Coulomb gas is already in the abstract of the first seminal article of Dyson[Dys62b] and of Ginibre [Gin65]. The terms Fermi gas and one-component plasma are also used.

2.4. The Wigner jellium and electrons in metals. It turns out that Coulomb gases are related toanother famous model of mathematical physics also due to Wigner. Let S ⊂ Rd be compact and let µbe a positive measure on Rd with µ(Rd) = α > 0. The Coulomb gas with potential

V =

{− 1

n Uµ on S,

+∞ outside S

is known as a Wigner jellium with background µ, and is said to be charge neutral when n = α. Thebackground µ models a positive charge α smeared out on supp(µ). This model, or more precisely itsthermodynamic limit as |S| → +∞, was derived by Wigner in 1938 as an approximation of the Hartree –Fock quantum model in order to model electrons in metals [Wig38], see also the 1904 pre-quantum workby Thomson [Tho04] on electrons and the structure of atoms. Conversely, a Coulomb gas with smoothpotential V can be seen as a jellium with background µ of density ∆V

cd. From this point of view, by looking

at (1.8), the complex Ginibre ensemble can be seen as a Jellium with full space Lebesgue background.The measure µ is positive when V is sub-harmonic (meaning that ∆V ≥ 0). If V is not sub-harmonicthen µ is no-longer positive but we may interpret it as an opposite charge on the subset {∆V < 0}. Werefer for instance to [CGZJ20a, CGZJ20b] for a bibliography and a discussion. The term jellium wasapparently coined by Conyers Herring, the smeared charge being viewed as a positive “jelly”, see [Hug06].

8 DJALIL CHAFAÏ

2.5. Random polynomials. The Coulomb or log gases emerging from random matrix theory describethe law of the eigenvalues of a random matrix, the roots of the characteristic polynomial. This randompolynomial has random dependent coefficients. We could study the distribution of the roots of randompolynomials with random independent coefficients. Actually this question emerged from various fields ofresearch including algebraic and geometric analysis and number theory, for instance with the works ofLittlewood and Offord in the 1920s, independently of the works of the statisticians on the spectral analysisof empirical covariance matrices. The simplest model that we could imagine is a random polynomial withindependent and identically distributed coefficients. This model is known as Kac polynomials, and thedistribution of the roots was computed in the Gaussian case by John Hammersley in 1956. There areseveral other natural models of random polynomials and plenty of works on such models. The gasesemerging from these models are two-dimensional but differ from Coulomb gases due to the presence inthe energy of an additional non-quadratic term with respect to the empirical measure. For more details,we refer for instance to [BBL96, BBLu92, KZ13, But17] and references therein. When the degree tendsto infinity, such models give rise to random analytic functions, see for instance [KZ14, HKPV09, But17]and references therein.

3. First order global asymptotics and large deviations

Let Pn be the Coulomb gas as in (2.2). If x1, . . . , xn are pairwise distinct elements of Rd, which holdsalmost everywhere with respect to Pn in (Rd)n, we get from (2.3) that

En(x1, . . . , xn) = n2E 6=V (µx1,...,xn)

where

E 6=V (µ) =

∫V dµ +

12

∫∫1u6=vg(u − v)dµ(u)dµ(v) and µx1,...,xn =

1n

n∑

i=1

δxi .

The probability measure Pn is exchangeable in the sense that it is invariant by permutation of the nparticles. The system is mean-field in the sense that each particle interacts with all the other particlesvia their empirical measure. The density of Pn at (x1, . . . , xn) is a function of µx1,...,xn and rewrites

exp(

− βn2E 6=V (µx1,...,xn)

)

Zn. (3.1)

In terms of asymptotic analysis, we expect that E 6=V ≈ EV as n → ∞, and the Laplace method suggests

that under Pn, the empirical measure µx1,...,xn concentrates as n → ∞ around the minimizers of EV .Since there is a unique minimizer known as the equilibrium measure µV , we expect that the empiricalmeasure µx1,...,xn under Pn converges towards µV as n → ∞. More precisely, for all n, let us define

Xn = (Xn,1, . . . , Xn,n) ∼ Pn and µn =1n

n∑

k=1

δXn,k.

3.1. The large deviations principle. For all Borel subsets A ⊂ M1(Rd), P(µn ∈ A) = Pn(µx1,...,xn ∈A). The following theorem translates mathematically the intuition above based on the Laplace principle:P(µn ∈ A) ≈n→∞ e−βn2 infA(EV −EV (µV )). The difficulty lies in the singularity of the Coulomb interaction.

Theorem 3.1 (Large deviations principle). We have

limn→∞

log Zn

βn2= −EV (µV ).

Moreover the sequence (µn)n satisfies to a large deviations principle of speed n2 and good rate function

EV − EV (µV ), in other words for all Borel subset of A ⊂ M1(Rd), we have

EV (µV ) − infint(A)

EV ≤ limn→∞

logP(µn ∈ A)βn2

≤ limn→∞

logP(µn ∈ A)βn2

≤ EV (µV ) − infclo(A)

EV

where int(A) and clo(A) are the interior and closure of A respectively.

About the proof. The first proof of such a result dates back to [BAG97] and concerns one-dimensionallog-gases. It is inspired by the work of Voiculescu on a Boltzmann point of view over free entropyand random matrices. Later contributions include [PH98, Har12, CGZ14]. The approach developed in[DLR20, Ber18b, Gar19] is very efficient. �

ASPECTS OF COULOMB GASES 9

Theorem 3.1 remains valid when β = βn provided that

limn→∞

nβn = +∞.

This can be called the “low temperature regime”. In the “high temperature regime” β = βn with

limn→∞

nβn = κ ∈ (0, +∞),

then Theorem 3.1 remains valid provided that we replace EV by the new functional

E +1κ

Entropy(· | νV,κ) = EV +1κ

Entropy(· | dx) + cV,κ

where νV,κ has density proportional to e−κV , and where Entropy is the Kullback – Leibler divergence orrelative entropy. Note that −Entropy(· | dx) is by definition the Boltzmann – Shannon entropy. We shouldalso replace µV in Theorem 3.1 by the minimizer of this new functional. This is also known as the crossoverregime, interpolating between µV and νV,κ. Formally, if we turn off the interaction by taking g = 0 andif we take βn = κ/n then Pn is the product probability measure ν⊗n

V,κ and the large deviations principlebecomes the classical Sanov theorem associated to the law of large numbers for independent randomvariables. The crossover regime is considered for instance in [CLMP92, BG99, Gar19, AB19, AS19].

3.2. First order global asymptotics. The (weak) convergence in M1(Rd) is metrized by the bounded-Lipschitz distance defined by

dBL(µ, ν) = sup{ ∫

fd(µ − ν) : ‖f‖∞ ≤ 1, ‖f‖Lip ≤ 1}

where the supremum runs over all measurable functions f : Rd → R and where

‖f‖∞ = supx

|f(x)| and ‖f‖Lip = supx 6=y

|f(x) − f(y)||x − y| .

Now for all r ≥ 0, by Theorem 3.1 with A = Ar = {µ ∈ M1(Rd) : dBL(µ, µV ) ≥ r}, for n large enough,

e−crβn2 ≤ P(dBL(µn, µV ) ≥ r) ≤ e−Crβn2

, (3.2)

where cr, Cr > 0 are constants depending on Ar and EV but not on n. In particular, for all ε > 0,∑

n

P(dBL(µn, µV ) > ε) < ∞.

By the Borel – Cantelli lemma, it follows that regardless of the way we choose a common probability spaceto define the sequence of random vectors (Xn)n, we have, almost surely,

limn→∞

dBL(µn, µV ) = 0. (3.3)

This is a sort of law of large numbers for our system of exchangeable particles. They are not independentdue to the Coulomb interaction, and the information about the interaction remains in µV . We refer to[Ser15, Ber18a, Gar19] for the relation to the notion of “Gamma convergence”.

3.3. Weakly confining versus strongly confining potential. We could say that V is weakly confiningwhen (1.10) holds, and that V is strongly confining when (1.13) holds. The integrability condition (2.1)may hold for weakly confining potentials. An example of a two dimensional Coulomb gas with a weaklyconfining potential is given by the Forrester – Krishnapur spherical ensemble considered in the sequel, forwhich the equilibrium measure is not compactly supported and is heavy-tailed.

3.4. Concentration of measure. The proof of Theorem 3.1 can be adapted in order to provide quan-titative (meaning non-asymptotic) estimates for deviation probabilities. Namely, for all r ≥ 0,

P(dBL(µn, µV ) ≥ r) =1

Zn

∫

dBL(µn,µV )≥r

e−βn2E 6=V

(µx1,...,xn )dx1 · · · dxn.

Now if we could approximate E 6=V (µx1,...,xn) with EV (µx1,...,xn) and use an inequality of the form

dBL(µ, µV ) ≤ c(EV (µ) − EV (µV )),

and use a bound of the form

log Zn ≥ n2EV (µV ) + n(βE(µV ) + cV ), (3.4)

10 DJALIL CHAFAÏ

then we would obtain a concentration of measure inequality of the form

P(d(µn, µV ) ≥ r) ≤ e−cn2r2+o(n2).

for all n and all r ≥ rn for some threshold rn. Actually the quantity EV (µx1,...,xn) is infinite due tothe atomic nature of µx1,...,xn and the method requires then a regularization procedure. The details arein [CHM18]. The method, inspired by [MMS14], is related to [RS16]. See also [GZ19a, MS19, Ber19a,PG20] for other variations on this topic. Moreover we could replace the bounded-Lipschitz distance by aKantorovich– Wasserstein distance, provided a growth assumption on V .

Such concentration inequalities around the equilibrium measure provide typically an upper bound onthe speed of the almost sure convergence. More precisely if rn is such that

∑n e−cn2r2

n+o(n2) < ∞, thenby the Borel – Cantelli lemma, we get that almost surely, for n large enough,

dBL(µn, µV ) ≤ rn.

On the other hand, in the case of one-dimensional log-gases with strongly convex potential V suchas the Gaussian unitary ensemble, another approach is possible for concentration of measure, related tologarithmic Sobolev inequalities, see for instance [CL20] and references therein.

4. Edge behavior

We suppose in this section that V is strongly confining, in particular the equilibrium measure µV iscompactly supported (Theorem 1.2). The convergence (3.3) holds in a weak sense, which does not implythe convergence of the support. The most general result about the convergence of the support is probably[Ame21], and appears as a refinement of [CHM18, Theorem 1.12]. When V is rotationally invariant, thisprovides constants c, r∗, p > 0 such that for all n and r ≥ r∗,

P

(max

1≤k≤n|Xn,k| ≥ r

)≤ e−cnrp

.

The fluctuation at the edge is a difficult subject which is well understood for one-dimensional log-gases,for which it gives rise to Tracy – Widom laws. For strongly confined rotationally invariant determinantaltwo-dimensional Coulomb gases, it gives rise to Gumbel laws. An explicit analysis of the Ginibre Coulombgas is presented in the sequel (Theorem 7.14), see also [CP14, JQ17, CGZJ20b, Seo20, Ame21, BGZ18,GZ18, BGZNW21, CGZJ20a] for more results in the same spirit.

5. Global fluctuations and Gaussian free field

Formally, from (1.9) we could write

E(µ) =12

〈−c−1d ∆Uµ, Uµ〉 + 〈−c−1

d ∆V, Uµ〉,

and thus

βn2E(µ) =12

〈−βc−1d ∆Unµ, Unµ〉 + 〈−nβc−1

d ∆V, Unµ〉.In view of the Coulomb gas formula (3.1), this suggests to interpret as n → ∞ the random functionUnµx1,...,xn

under Pn as a Gaussian with covariance operator K = cd(−β∆)−1. Actually such an objectis known as a Gaussian Free Field (GFF). Next, again from (1.9), this suggests to interpret formallyas n → ∞ the random measure nµx1,...,xn = −c−1

d ∆Uµx1,...,xn= AUµx1,...,xn

under Pn as a Gaussianrandom measure with covariance operator A2K = (−c−1

d ∆)2(cd(−β∆)−1) = −(βcd)−1∆. This argumentwould involve in principle a change of variable and a Jacobian, that we do not consider here. This leadsnaturally to conjecture that for a smooth enough test function f : Rd → R,

n( ∫

fdµn − E

∫fdµn

)=

n∑

k=1

f(Xn,k) − E(f(Xn,k)) d−→n→∞

N(

0,1

βcd

∫|∇f |2dx

). (5.1)

This can be seen as the “central limit theorem” statement associated to the “law of large numbers”statement (3.3). The limiting variance could be perturbed by edge effects depending on the relativeposition of the support and regularity of f and µV . We could have also an additional bias correction.

The Coulomb interaction together with the confinement produces a rigidity of the global configurationand reduces the variance of linear statistics. Indeed (5.1) comes with an n scaling that differs from theusual

√n scaling for independent random variables (no interaction).

ASPECTS OF COULOMB GASES 11

The covariance of the limiting Gaussian in (5.1) is easily guessed from the Hessian at the minimizer ofthe rate function in the large deviations principle of Theorem 3.1. This CLT – LDP link is well known.The GFF is an example of a log-correlated Gaussian field [DRSV17], a fashionable subject.

A statement similar to (5.1) is proved rigorously in [RV07] for the complex Ginibre ensemble by usingits exact solvability (determinantal structure). See also [AHM15, AHM11]. Extensions to non-exactlysolvable two-dimensional Coulomb gases are considered in [BBNY19, LS18, Ser20a, LZ20].

For one-dimensional log-gases emerging from random matrix theory, central limit theorems such as(5.1) were established using the Laplace transform and “loop equations” in [Joh98]. See also [PS11,BGG17, BLS18, HL21] and references therein for extensions and generalizations.

6. Aspects of general exact solvability

Theorem 6.1 is taken from [Cha19a] and [CL20] (see also [CFS21]).

Theorem 6.1 (Exact distributions for special linear statistics of general gases). Let X = (X1, . . . , Xn)be a random vector of (Rd)n, n, d ≥ 1, with density proportional to

e−∑

iV (xi)

∏

i<j

W (xi − xj)

where V : Rd → [0, +∞] and W : Rd → [0, +∞] are measurable.

• If V and W are homogeneous in the sense that for some a, b ≥ 0, and for all λ ≥ 0 and x ∈ Rd,

V (λx) = λaV (x) and W (λx) = λbW (x),

then

V (X1) + · · · + V (Xn) ∼ Gamma(nd

a+

n(n − 1)b2a

, 1)

.

• If V = γ |·|2 for some γ > 0 then

X1 + · · · + Xn ∼ N(

0,n

2γId

),

and moreover the orthogonal projection π on the subspace {(z, . . . , z) : z ∈ Rd} of (Rd)n satisfiesπ(X) = X1+···+Xn

n (1, . . . , 1), and furthermore π(X) and π⊥(X) = X − π(X) are independent.

Proof of Theorem 6.1. Recall that linear change of variable is valid for integrals of measurable functions.First formula. For all θ > 0, we have, with the substitution xi =

(1

1+θ

)1/ayi,

∫

(Rd)n

e−θ∑

iV (xi)e−

∑i

V (xi)∏

i<j

W (xi −xj)dx =( 1

1 + θ

) nda +

(n2−n)2

b2a

∫

(Rd)n

e−∑

iV (yi)

∏

i<j

W (yi −yj)dy.

We recognize the Laplace transform of Gamma(

nda + β n(n−1)b

2a , 1), namely

∫ ∞

0

e−θxxα−1e−λxdx =∫ ∞

0

xα−1e−(λ+θ)xdx =( λ

λ + θ

)α Γ(α)λα

,

therefore∑

i V (Xi) ∼ Gamma(

nda + n(n−1)b

2a , 1).

Second formula. For all θ ∈ Rd, we have, with the substitution yi = xi + 12γ θ (a translation or shift),

∫

(Rd)n

e−θ·∑

ixie−

∑i

V (xi)∏

i<j

W (xi − xj)dx = en

2γ|θ|2

2

∫

(Rd)n

e−∑

iV (yi)

∏

i<j

W (yi − yj)dy,

and we recognize the Laplace transform of the Gaussian law N(0, n

2γ Id

). Finally the properties related

to π(X) follow from the quadratic nature of V and the shift invariance of W , and correspond to afactorization of the law of X , namely, denoting π⊥(x) = x − π(x) and using |x|2 = |π(x)|2 + |π⊥(x)|2(Pythagoras theorem) and xi − xj = π⊥(x)i − π⊥(x)j (from the definition of π), we get

e−∑

iV (xi)

∏

i<j

W (xi − xj) = e−γ|π(x)|2 × e−γ|π⊥(x)|2 ∏

i<j

W (π⊥(x)i − π⊥(x)j).

This provides the independence of π(X) and π⊥(X) as well as the fact that π(X) ∼ N (0, 12γ Id). �

12 DJALIL CHAFAÏ

Corollary 6.2 (Exact laws for beta-Ginibre gases). Let us consider Xn = (Xn,1, . . . , Xn,n) ∼ Pn where

Pn is as in (2.4) with β > 0 and V = 12 |·|2. In other words, the density of Pn in Cn is given by

(z1, . . . , zn) ∈ Cn 7→ e−n β

2 (|z1|2+···+|zn|2)

Zn

∏

i<j

|zi − zj|β .

Then

Xn,1 + · · · + Xn,n ∼ N(

0,I2

β

)and |Xn|2 = |Xn,1|2 + · · · + |Xn,n|2 ∼ Gamma

(n + β

n(n − 1)4

, βn

2

),

and in particular

E(|Xn,1 + · · · + Xn,n|2) =2β

and E(|Xn|2) = E(|Xn,1|2 + · · · + |Xn,n|2) =2β

+n − 1

2.

When β = 2 we recover the Ginibre gas (2.5). Beyond this case, and up to our knowledge, it seemsthat there is no useful matrix model with independent entries for which the spectrum follows this β gas.

With β = 1n , we get as n → ∞ that the variance of the Gauss – Ginibre crossover is 2 + 1

2 = 52 .

Proof of Corollary 6.2. It suffices to use Theorem 6.1 with d = 2, V = n β2 |·|2, W = |·|β, for which a = 2

and b = β, and the scaling property σZ ∼ Gamma(α, λ

σ

)when Z ∼ Gamma(α, λ), for any σ > 0.

Note that in the determinantal case β = 2, Theorem 7.13 gives that n|X |2 ∼ Gamma(1+2+ · · ·+n, 1)since it has the law of a sum of n independent random variables of law Gamma(1, 1), . . . , Gamma(n, 1). �

Remark 6.3 (Real case). For all β > 0, n ≥ 2, let Pn be the law on Rn with density

(x1, . . . , xn) ∈ Rn 7→ e−n β

4 (x21+···+x2

n)

Zn

∏

i<j

|xi − xj |β .

The normalization Zn can be explicitly computed via a Mehta – Selberg integral [FW08]. It is a quadrati-cally confined one-dimensional log-gas known as the real beta Hermite gas. The case β = 2 correspondsto GUE. If Xn = (Xn,1, . . . , Xn,n) ∼ Pn, then the proof of Theorem 6.2 provides

Xn,1 + · · · + Xn,n = N(

0,2β

)and X2

n,1 + · · · + X2n,n ∼ Gamma

(n

2+

βn(n − 1)4

,βn

4

),

and in particular,

E((Xn,1 + · · · + Xn,n)2) =2β

and E(X2n,1 + · · · + X2

n,n) =2β

+ n − 1.

Alternatively, these formulas can also be derived by using the tridiagonal random matrix model of Dumitriuand Edelman [DE02] valid for all real beta Hermite gases, see for instance [CL20].

Remark 6.4 (Langevin dynamics). The Boltzmann – Gibbs measure Pn defined in (2.2) is the invariantlaw of the Kolmogorov diffusion process (Xt)t≥0 solution of the stochastic differential equation

dXt =√

2α

βdBt − α∇En(Xt)dt (6.1)

where (Bt)t≥0 is a standard Brownian motion on (Rd)n, and where α > 0 is an arbitrary parameterwhich corresponds to a deterministic time change. The infinitesimal generator of the associated Markovsemi-group is the second order linear differential operator without constant term

L = α( 1

β∆ − ∇En · ∇

). (6.2)

The most standard parametrizations are α = 1, which allows to interpret 1/β as the temperature of theBrownian part, and α = β. See for instance [Cha15, Roy07, BGL14]. Since En is a two-body interactionenergy, the operator L can be seen as a mean-field particle approximation of a McKean – Vlasov dynamics,see for instance [BCF18, Ser20b]. The singularity of g makes non-obvious the well-posedness or absence ofexplosion of (6.1), and we refer to [RS93, AGZ10, EY17] for one-dimensional log-gases, and to [BCF18]for the (two-dimensional) beta-Ginibre gas. See also [BD20] and [AB19] for more recent results on suchdynamics. Historically (6.1) emerges as the description of the dynamics of the spectrum of HermitianOrnstein – Uhlenbeck processes, and is nowadays called a Dyson process, named after [Dys62a]. We say

ASPECTS OF COULOMB GASES 13

that (6.1) is a gradient dynamics because the drift is the gradient of a function. From the point of view ofstatistical physics a stochastic differential equation such as (6.1) is also known as an overdamped Langevindynamics, which is a degenerate version of the true (kinetic under-damped) Langevin dynamics, see forinstance [CF19]. Langevin dynamics can be used for the numerical simulation of Pn, see for instance[CF19, CFS21] and references therein. Dynamics such as (6.1) can be used as an interpolation devicebetween X0 and X∞ ∼ Pn, possibly by using conservation laws related to eigenfunctions. In this spirit,and following [BCF18, CL20], we could prove Corollary 6.2 by using the fact that

∑i xi and

∑i |xi|2 are

essentially eigenfunctions of (6.2), producing Ornstein – Uhlenbeck and Cox – Ingersoll – Ross processesfor which the targeted Gaussian and Gamma laws are invariant.

7. Exactly solvable two-dimensional gases from random matrix theory

The spectrum of several random matrix models are gases in dimension d ∈ {1, 2} with W = g. Thecases β ∈ {1, 2, 4} play often a special role related to algebra. We refer to [Meh04, ER05, For10] for moredetails on the zoology of random matrices. It is natural to ask if there exists a random matrix modelwith independent entries for which the spectrum is distributed according to the beta gas (2.4). Up toour knowledge, the answer is negative for (2.4) in general but positive for the Ginibre gas (2.5).

7.1. Ginibre model. A (complex) Ginibre random matrix M is an n × n complex matrix such that{

ℜMi,j , ℑMi,j : 1 ≤ i, j ≤ n}

(7.1)

are independent and Gaussian random variables of law N (0, 12 ). In other words the complex random

variables {Mi,j : 1 ≤ i, j ≤ n} are independent and Gaussian of law N (0, 12 I2). Note that E(|Mi,j |2) = 1.

Let (λ1, . . . , λn) be the eigenvalues of M seen as an exchangeable random vector of Cn. This meansthat we randomize the numbering of the eigenvalues with an independent uniform random permutationof {1, . . . , n}. Equivalently, this corresponds to consider the random multi-set encoding the spectrum,keeping by this way the possible multiplicities but discarding the numbering of the eigenvalues.

Theorem 7.1 (From the Ginibre random matrix to the Ginibre gas). The exchangeable random vector(

λ1√n

, . . . ,λn√

n

)

is distributed according to the Ginibre gas (2.5). In other words (λ1, . . . , λn) has density ϕn as in (2.5).

Idea of the proof. The set of n×n complex matrices with multiple eigenvalues has zero Lebesgue measure.Since the law of M is absolutely continuous, it follows that almost surely M is diagonalizable with distincteigenvalues. The density is proportional to (M∗ = M

⊤is the conjugate-transpose of M)

M 7→ e−∑n

i,j=1|Mi,j |2

= e−Trace(MM∗).

In order to compute the law of the spectrum of M, an idea is to use for instance the Schur unitarydecomposition as a change of variable. Namely, if M is diagonalizable, then the Schur decomposition isthe matrix factorization M = U(D + N)U∗ where U is unitary, D is diagonal, and N is upper triangularwith null diagonal (nilpotent). The matrix D carries the eigenvalues of M . We have the decoupling

Trace(MM∗) = Trace(DD∗) + Trace(NN∗).

This allows to integrate out (N, U) in the density and to get that the law of the eigenvalues of M is givenby (2.5). The term

∏i<j |xi − xj |2 is the modulus of the determinant of the Jacobian of the change of

variable. We obtain that for every symmetric bounded (or positive) measurable function F : Cn → R,

E[F (λ1, . . . , λn)] =∫

Cn

F (z1, . . . , zn)ϕn(z1, . . . , zn)dz1 · · · dzn

where dz1 · · · dzn stands for the Lebesgue measure on Cn = R

2n. The result goes back to [Gin65]. Thescheme of proof that we follow here can be found in [KS11], see also [Meh04, For10, Ch. 15]. �

Remark 7.2 (Immediate properties of Ginibre random matrices).• Since the law of M is absolutely continuous, almost surely MM∗ 6= M∗M (non-normality);• By the law of large numbers, almost surely, as n → ∞, 1√

nM has orthonormal rows/columns;

• The law of M is bi-unitary invariant: if U and V are unitary then UMV and M have same law;

14 DJALIL CHAFAÏ

• The Hermitian random matrices 1√2(M + M∗) and 1√

2i(M − M∗), the matrix real and imaginary

parts of M, are independent and belong to the Gaussian Unitary Ensemble (GUE): their density

is proportional to H 7→ e− 12 Trace(H2); Conversely, if H1 and H2 are independent copies of the

Gaussian Unitary Ensemble then the random matrices 1√2(H1 + iH2) and M have same law.

The exact solvability of the Ginibre gas (2.5) is largely due to a determinantal structure studied below,itself related to the fact that β = 2 and W = g. More precisely, first of all, from (2.5) we have

ϕn(z1, . . . , zn) =∏n

k=1 γ(zk)∏nk=1 k!

∏

i<j

|zi − zj|2 (7.2)

where γ is the density of N (0, 12 I2) given for all z ∈ C by

γ(z) =e−|z|2

π.

Theorem 7.3 (Determinantal structure and marginals). For all n ≥ 1 and (z1, . . . , zn) ∈ Cn,

ϕn(z1, . . . , zn) =1n!

det [Kn(zi, zj)]1≤i,j≤n

where the kernel Kn is given for all z, w ∈ C by

Kn(z, w) =√

γ(z)γ(w)n−1∑

ℓ=0

(zw)ℓ

ℓ!.

More generally, for all 1 ≤ k ≤ n, the marginal density

(z1, . . . , zk) ∈ Ck 7→ ϕn,k(z1, . . . , zk) =

∫

Cn−k

ϕn(z1, . . . , zn)dzk+1 · · · dzn

satisfies, for all (z1, . . . , zk) ∈ Ck,

ϕn,k(z1, . . . , zk) =(n − k)!

n!det [Kn(zi, zj)]1≤i,j≤k .

In particular for k = n we get ϕn,n = ϕn, while for k = 1 we get, for all z ∈ C,

ϕn,1(z) =γ(z)

n

n−1∑

ℓ=0

|z|2ℓ

ℓ!.

We say that the spectrum of M is a Gaussian determinantal point process, see [HKPV09, Ch. 4].The “k-point correlation” is Rn,k(z1, . . . , zk) = n!

(n−k)! ϕn,k(z1, . . . , zk) = det[Kn(zi, zj)]1≤i,j≤k.

Idea of proof. Following for instance [Meh04, Sec. 5.2 and Ch. 15], we get, starting with (7.2),

ϕn(z1, . . . , zn) =∏n

k=1 γ(zk)∏nk=1 k!

∏

1≤i<j≤n

(zi − zj)∏

1≤i<j≤n

(zi − zj)

=∏n

k=1 γ(xk)n!

det[ zi−1

j√(i − 1)!

]1≤i,j≤n

det[ zj

i−1

√(i − 1)!

]1≤i,j≤n

=1n!

det[Kn(zi, zj)

]1≤i,j≤n

.

On the other hand, the orthogonality of { zℓ√

ℓ!: 0 ≤ ℓ ≤ n − 1} in L2(C, γ) gives the identities

∫

C

Kn(x, x)dx = n and∫

C

Kn(x, y)Kn(y, z)dy = Kn(x, z), x, z ∈ C.

Finally the formula for ϕn,k follows by expanding the determinant in ϕn and using these identities. �

Theorem 7.4 (Mean circular Law). Let λ1, . . . , λn be as in Theorem 7.1 and let us define

µn =1n

n∑

k=1

δ λk√n

.

ASPECTS OF COULOMB GASES 15

Let µ∞ be the uniform distribution on the unit disc {z ∈ C : |z| ≤ 1} with density z ∈ C 7→ 1|z|≤1

π . Then

EµnCb−→

n→∞µ∞.

Proof. Let ϕn,1 be as in Theorem 7.3. For all f ∈ Cb(C,R), we have, using Theorem 7.1,

E

∫fdµn =

1n

n∑

k=1

∫

Cn

f( zk√

n

)ϕn(z1, . . . , zn)dz1 · · · dzn = n

∫

C

f(z)ϕn,1(√

nz)dz.

Thus Eµn has density nϕn,1(√

n•). By Theorem 7.3 and Lemma 7.5, if K ⊂ {z ∈ C : |z| 6= 1} is compact,

limn→∞

supz∈K

∣∣∣nϕn,1(√

nz) − 1|z|≤1

π

∣∣∣ =1π

limn→∞

supz∈K

∣∣∣e−n|z|2

en(n|z|2) − 1|z|≤1

∣∣∣ = 0.

It follows then by dominated convergence that EµnCb−→

n→∞µ∞. �

Lemma 7.5 (Exponential series). For every n ≥ 1 and z ∈ C,

|en(nz) − enz1|z|≤1| ≤ rn(z)

where en(z) =∑n−1

ℓ=0zℓ

ℓ! is the truncated exponential series and

rn(z) =en

√2πn

|z|n( n + 1

n(1 − |z|) + 11|z|≤1 +

n

n(|z| − 1) + 11|z|>1

).

Proof of Lemma 7.5. As in Mehta [Meh04, Ch. 15], for every n ≥ 1, z ∈ C, if |z| ≤ n then

∣∣∣ez − en(z)∣∣∣ =

∣∣∣∞∑

ℓ=n

zℓ

ℓ!

∣∣∣ ≤ |z|nn!

∞∑

ℓ=0

|z|ℓ(n + 1)ℓ

=|z|nn!

n + 1n + 1 − |z| ,

while if |z| > n then

|en(z)| ≤n−1∑

ℓ=0

|z|ℓℓ!

≤ |z|n−1

(n − 1)!

n−1∑

ℓ=0

(n − 1)ℓ

|z|ℓ ≤ |z|n−1

(n − 1)!|z|

|z| − n + 1.

Therefore, for every n ≥ 1 and z ∈ C,

|en(nz) − enz1|z|≤1| ≤ nn

n!

(|z|n n + 1

n + 1 − |nz|1|z|≤1 + |z|n−1 |nz||nz| − n + 1

1|z|>1

).

It remains to use the Stirling bound√

2πnnn ≤ n!en to get the first result. �

Remark 7.6 (Probabilistic view). There is a probabilistic interpretation of Lemma 7.5. For all z ∈ C,

limn→∞

nϕn,1(√

nz) =1π

(1|z|<1 +

12

1|z|=1

).

Namely, by rotational invariance, it suffices to consider the case z = r > 0. Next, if Y1, . . . , Yn areindependent and identically distributed random variables following the Poisson law of mean r2, then

e−nr2

en(nr2) = P(Y1 + · · · + Yn < n) = P

(Y1 + · · · + Yn

n< 1

).

Now limn→∞Y1+···+Yn

n = r2 almost surely by the strong law of large numbers, and thus the probability inthe right-hand side above tends as n → ∞ to 0 if r > 1 and to 1 if r < 1. In other words, for all r 6= 1,

limn→∞

e−nr2

en(nr2) = 1r<1.

It remains to note that for r = 1 by the central limit theorem we get

P

(Y1 + · · · + Yn

n< 1

)= P

(Y1 + · · · + Yn − n√n

< 0)

−→n→∞

12

.

16 DJALIL CHAFAÏ

Remark 7.7 (Incomplete gamma function). It is well known that the Gamma and the Poisson laws areconnected. Namely, if X ∼ Gamma(n, λ) with n ≥ 1 and λ > 0 and Y ∼ Poisson(r) with r > 0 then

P(X ≥ λr) =1

(n − 1)!

∫ ∞

r

xn−1e−xdx = e−rn−1∑

ℓ=0

rℓ

ℓ!= P(Y ≥ n).

Also we could use Gamma random variables instead of Poisson random variables in Remark 7.6. Notealso that the integral in the middle of the formula above is the incomplete Gamma function Γ(n, r). Thisallows to benefit from the asymptotic analysis of this special function, see [KS11] and references therein.

Theorem 7.8 (Strong circular law). With the notations of Theorem 7.4, almost surely,

µnCb−→

n→∞µ∞.

Note that this convergence holds regardless of the way we define the random matrices on the sameprobability space when n varies. This is an instance of the concept of complete convergence, see [Yuk98].

Idea of the proof. The argument, due to Jack Silverstein, is in [Hwa86]. It is similar to the quick proofof the strong law of large numbers for independent random variables with bounded fourth moment. Itsuffices to establish the result for an arbitrary compactly supported f ∈ Cb(C,R). Let us define

Sn =∫

C

f dµn and S∞ =1π

∫

|z|≤1

f(z)dz.

Suppose for now that we have

E[(Sn − ESn)4] = O( 1

n2

). (7.3)

By monotone convergence or by the Fubini – Tonelli theorem,

E

∞∑

n=1

(Sn − ESn)4 =∞∑

n=1

E[(Sn − ESn)4] < ∞

and thus∑∞

n=1 (Sn − ESn)4< ∞ almost surely, which implies limn→∞ Sn −ESn = 0 almost surely. Since

limn→∞ ESn = S∞ by Theorem 7.4, we get that almost surely

limn→∞

Sn = S∞.

Finally, one can swap the universal quantifiers on ω and f thanks to the separability of the set of compactlysupported continuous bounded functions C → R equipped with the supremum norm. To establish thefourth moment bound (7.3), we set

Sn − ESn =1n

n∑

k=1

Zk with Zk = f(

λk√n

)− Ef

(λk√

n

).

Next, we obtain, with∑

k1,... running over distinct indices in 1, . . . , n,

E

[(Sn − ESn)4

]=

1n4

∑

k1

E[Z4k1

]

+4n4

∑

k1,k2

E[Zk1 Z3k2

]

+3n4

∑

k1,k2

E[Z2k1

Z2k2

]

+6n4

∑

k1,k2,k3

E[Zk1 Zk2 Z2k3

]

+1n4

∑

k1,k2,k3,k3,k4

E[Zk1 Zk3 Zk3Zk4 ].

The first three terms of the right are O(n−2) since max1≤k≤n |Zk| ≤ ‖f‖∞. The expressions of ϕn,3 andϕn,4 from Theorem 7.3 allow to show that the remaining two terms are also O(n−2), see [Hwa86]. �

The following theorem includes Theorem 7.4, which corresponds to the case k = 1.

ASPECTS OF COULOMB GASES 17

Theorem 7.9 (Chaoticity). Let µ∞ be the uniform distribution on the unit disc {z ∈ C : |z| ≤ 1}. Forall 1 ≤ k ≤ n, denoting by Pn,k the k-dimensional marginal distribution of the Ginibre gas (2.5), we have

Pn,kCb−→

n→∞µ⊗k

∞ .

Idea of the proof. The measures Pn,k and µ∞ have densities ϕn,k and z ∈ C 7→ ϕ∞(z) = π−11|z|≤1.

The case k = 1 is nothing else but Theorem 7.4, namely Pn,1Cb−→

n→∞µ∞. This comes via dominated

convergence from the fact that limn→∞ ϕn,k = ϕ∞ uniformly on compact subsets of {z ∈ C : |z| 6= 1}.Let us consider now the case k = 2. Here again, by dominated convergence, it suffices to show that

limn→∞

ϕn,2 = ϕ⊗2∞

uniformly on compact subsets of {(z1, z2) ∈ C2 : |z1| 6= 1, |z2| 6= 1, z1 6= z2}.By Theorem 7.3, for all z1, z2 ∈ C,

ϕn,2(z1, z2) =n

n − 1e−n(|z1|2+|z2|2)

π2

(en(n|z1|2)en(n|z2|2) − |en(nz1z2)|2

)

=n

n − 1ϕn,1(z1)ϕn,1(z2) − n

n − 1e−n(|z1|2+|z2|2)

π2|en(nz1z2)|2 (7.4)

where en is as in Lemma 7.5. It follows that for any n ≥ 2 and z1, z2 ∈ C,

∆n(z1, z2) = ϕn,2(z1, z2) − ϕn,1(z1)ϕn,1(z2)

=1

n − 1ϕn,1(z1)ϕn,1(z2) − n

n − 1e−n(|z1|2+|z2|2)

π2|en(nz1z2)|2. (7.5)

In particular, using ϕn,2 ≥ 0 for the lower bound,

−ϕn,1(z1)ϕn,1(z2) ≤ ∆n(z1, z2) ≤ 1n − 1

ϕn,1(z1)ϕn,1(z2).

From this and Lemma 7.5 we first deduce that for any compact subset K of {z ∈ C : |z| > 1}

limn→∞

supz1∈C

z2∈K

|∆n(z1, z2)| = limn→∞

supz1∈Kz2∈C

|∆n(z1, z2)| = 0.

It remains to show that ∆n(z1, z2) → 0 as n → ∞ when z1 and z2 are in compact subsets of {(z1, z2) ∈C2 : |z1| < 1, |z2| < 1}. In this case |z1z2| ≤ 1, and Lemma 7.5 gives

|en(nz1z2)|2 ≤ 2e2nℜ(z1z2) + 2r2n(z1z2).

Next, using the elementary identity 2ℜ(z1z2) = |z1|2 + |z2|2 − |z1 − z2|2, we get

e−n(|z1|2+|z2|2)|en(nz1z2)|2 ≤ 2e−n|z1−z2|2

+ 2e−n(|z1|2+|z2|2)r2n(z1z2). (7.6)

Since |z1z2| ≤ 1, the formula for rn in Lemma 7.5 gives

e−n(|z1|2+|z2|2)r2n(z1z2) ≤ e−n(|z1|2+|z2|2−2−log |z1|2−log |z2|2) (n + 1)2

2πn.

Using (7.5), (7.6) and the bounds ϕn,1 ≤ π−1 and u − 1 − log u > 0 for 0 < u < 1, it follows that∆n(z1, z2) tends to 0 as n → ∞ uniformly in z1, z2 on compact subsets of

{(z1, z2) ∈ C2 : |z1| < 1, |z2| < 1, z1 6= z2}.

This finishes the proof of the case k = 2. The case k ≥ 3 follows from the case k = 2 by Lemma 7.11. �

Remark 7.10 (Impossibility of global uniform convergence of densities). The convergence of ϕn,1 cannothold uniformly on arbitrary compact sets of C since the point-wise limit is not continuous on the unitcircle. Similarly, the convergence of ϕn,2 cannot hold on {(z, z) : z ∈ C, |z| < 1} since ϕn,2(z, z) = 0 forany n ≥ 2 and z ∈ C while ϕ∞(z)ϕ∞(z) = π−2 6= 0 when |z| < 1.

18 DJALIL CHAFAÏ

Lemma 7.11 (Chaoticity). Let E be a Polish space. For all n ≥ 1, let Pn ∈ M1(En) be exchangeable,and for all 1 ≤ k ≤ n, let Pn,k ∈ M1(Ek) be its k-dimensional marginal distribution. For all n ≥ 1, letus pick a random vector Xn = (Xn,1, . . . , Xn,n) ∼ Pn and let us define the random empirical measure

µn =1n

n∑

i=1

δXn,i .

For all µ ∈ M1(E), the following properties are equivalent:

(1) µnCb−→

n→∞δµ in M1(M1(E)) (here we see µn a random variable taking values in M1(E));

(2) Pn,kCb−→

n→∞µ⊗k for any fixed k ≥ 1 (note that Pn,k has a meaning as soon as n ≥ k);

(3) Pn,2Cb−→

n→∞µ⊗2;

where these weak convergences are with respect to continuous and bounded test functions.

Proof. Folkloric in the domain of mean field particle systems. We refer to [BCF18] and references therein.�

Theorem 7.12 (Central limit phenomenon). Let µn be as in Theorem 7.4. Then, for all measurablef : C → R which are C1 in a neighborhood of the unit disc D = {z ∈ C : |z| ≤ 1} of the complex plane,

n[ ∫

fdµn − E

∫fdµn

]=

n∑

k=1

[f

(λk√

n

)− Ef

(λk√

n

)]d−→

n→∞N

( 14π

‖f‖2H1(D) +

12

‖f‖2H1/2(∂D)

)

where

‖f‖2H1(D) =

∫

D

|∇f |2dz and ‖f‖2H1/2(∂D) =

∑

k∈Z

|k||f̂(k)|2

where f̂(k) is the k-th Fourier coefficient of f on ∂D = {z ∈ C : |z| = 1}, namely

f̂(k) =1

2π

∫ 2π

0

f(eiθ)e−ikθdθ.

Note that ‖f‖H1/2(∂D) = 0 if f is analytic on a neighborhood of ∂D.

About the proof. It is known [CL95] that the cumulants of linear statistics of determinantal processeshave a nice form that can be used to prove a central limit theorem. This idea is followed in [RV07] inorder to produce the result, via combinatorial identities, and via reduction to polynomial test functions.The method is used for more general two-dimensional determinantal gases in [AHM11, AHM15]. �

Theorem 7.13 (Distribution of the moduli in the Ginibre model). Let (λ1, . . . , λn) be the exchangeablerandom vector considered in Theorem 7.1. Then the following equality in distribution holds

(|λ1|, . . . , |λn|) d= (Zσ(1), . . . , Zσ(n))

where Z1, . . . , Zn are independent non-negative random variables with123

Z2k ∼ Gamma(k, 1), 1 ≤ k ≤ n,

and where σ is a uniform random permutation of {1, . . . , n} independent of Z1, . . . , Zn. Equivalently, forall symmetric bounded measurable F : Rn → R, we have E(F (|λ1|, . . . , |λn|)) = E(F (Z1, . . . , Zn)).

This is an equality between two exchangeable laws on Rn, in other words an equality in law betweentwo configurations of unlabeled random points in R (multi-sets). Note in particular that for all 1 ≤ k ≤ n,taking F (x1, . . . , xn) =

∑i1,...,ik

distinctf(xi1 ) · · · f(xik

) gives equality of k-point correlation functions.

1The law Gamma(a, λ) has density x ∈ R 7→ λa

Γ(a)xa−1e−λx1x≥0, and Gamma(a, λ) ∗ Gamma(b, λ) = Gamma(a + b, λ).

2Note that (√

2Zk)2 ∼ Gamma(k, 12

) = Exponential( 12

)∗k = χ2(2k) since χ2(n) = Gamma( n

2, 1

2) for all n ≥ 1.

3For n = 1 we recover the Box – Muller formula |X|2 ∼ χ2(2) = Gamma(1, 12

) = Exponential( 12

) with X ∼ N (0, I2).

ASPECTS OF COULOMB GASES 19

Proof. From Theorem 7.1, the exchangeable random vector (λ1, . . . , λn) has density ϕn. It follows thatthe density of the exchangeable random vector (|λ1|, . . . , |λn|) is obtained from ϕn by integrating thephases in polar coordinates. In polar coordinates xk = rkeiθk , the density ϕn writes

(r1, . . . , rn, θ1, . . . , θn) 7→ e−∑

n

j=1r2

j

πn∏n

k=1 k!

∏

j<k

|rjeiθj − rkeiθk |2.

Now we have, denoting Σn the symmetric group of permutations of {1, . . . , n},∏

j<k

|rjeiθj − rkeiθk |2 =∏

j<k

(rjeiθj − rkeiθk)∏

j<k

(rjeiθj − rkeiθk)

= det[rk−1

j ei(k−1)θj

]1≤j,k≤n

det[rk−1

j e−i(k−1)θj

]1≤j,k≤n

=( ∑

σ∈Σn

(−1)sign(σ)n∏

j=1

rσ(j)−1j ei(σ(j)−1)θj

)( ∑

σ′∈Σn

(−1)sign(σ′)n∏

j=1

rσ′(j)−1j e−i(σ′(j)−1)θj

)

=∑

σ,σ′∈Σn

(−1)sign(σ)+sign(σ′)n∏

j=1

rσ(j)+σ′(j)−2j ei((σ(j)−σ′(j))θj).

If we integrate the phases, we note that only the terms with σ = σ′ contribute to the result, namely∫ 2π

0

· · ·∫ 2π

0

∏

j<k

|rjeiθj − rkeiθk |2dθ1 · · · dθn = (2π)n∑

σ∈Σn

n∏

j=1

r2(σ(j)−1)j = (2π)nper

[r2k

j

]1≤j,k≤n

where “per” stands for “permanent”. Therefore, the (exchangeable) density of the moduli is given by∫ 2π

0

· · ·∫ 2π

0

e−∑

n

j=1r2

j

πn∏n

k=1 k!

∏

j<k

|rjeiθj − rkeiθk |2dθ1 · · · dθn = perm[ 2

k!r2k

j e−r2j

]1≤j,k≤n

.

But if f1, . . . , fn : R → R are probability density functions then (x1, . . . , xn) 7→ perm[fj(xk)]1≤j,k≤n is thedensity of the random vector (Xσ(1), . . . , Xσ(n)) where X1, . . . , Xn are independent real random variableswith densities f1, . . . , fn and where σ is a uniform random permutation of {1, . . . , n} independent ofX1, . . . , Xn. Also the desired result follows from the formula above and the fact that for all 1 ≤ k ≤ n, anon-negative random variable Zk has density r 7→ 2

k! r2ke−r2

1r≥0 if and only if Z2k ∼ Gamma(k, 1).

This proof, essentially due to Kostlan [Kos92], see also [Rid03], relies on the determinantal nature ofϕn in (2.5), and remains usable for general determinantal processes, see for instance [HKPV09]. �

Theorem 7.14 (Spectral radius). With the notation of Theorem 7.1, almost surely

ρn = max1≤k≤n

|λk|√n

−→n→∞

1.

Moreover, denoting κn = log n2π − 2 log(log(n)),

√4nκn

(ρn − 1 −

√κn

4n

)d−→

n→∞Gumbel4.

Note the the second statement (Gumbel fluctuation) implies that ρn −→n→∞

1 in probability.

Regarding the convergence, see [BCGZ20] for a random analytic function point of view, related tothe central limit theorem. Regarding the fluctuation, see [Joh07, Ben10] for an interpolation with theTracy – Widom fluctuation at the edge of GUE.

Idea of proof. By Theorem 7.13 Since Z2k

d= E1+· · ·+Ek where E1, . . . , Ek are independent and identically

distributed exponential random variables of unit mean, we get, for every r > 0,

P(ρn ≤ √nr) =

∏

1≤k≤n

P

(E1 + · · · + Ek

n≤ r2

).

By the law of large numbers, this tends as n → ∞ to 0 or 1 depending on the position of r withrespect to 1. Moreover the central limit theorem suggests that ρn behaves as n → ∞ as the maximum of

4If X ∼ Exp(1) then − log(X) has Gumbel law and cumulative distribution function P(− log(X) ≤ x) = e−e−x.

20 DJALIL CHAFAÏ

independent and identically distributed Gaussian random variables, a situation for which it is known thatthe fluctuation follows the Gumbel law. The full proof is in [Rid03] and involves crucially a quantitativecentral limit theorem and the Borel – Cantelli lemma. The approach is robust and remains valid beyondthe Ginibre gas, for determinantal gases, see for instance [CP14, JQ17, GZ18] and references therein. �

Remark 7.15 (Real or quatertionic Ginibre model). How about an analogue of Theorem 7.1 when theentries of M are real Gaussian or real quaternionic Gaussian instead of complex Gaussian? Some answersare already in [Gin65]. In these cases, the density of the eigenvalues can be computed but it is not thebeta gas (2.4) with β ∈ {1, 4}. This is in contrast with the G(O|U|S)E triplet of the Hermtian randommatrix Dysonian universe [Dys62c]. See for instance [Ede97, Dub18b] and references therein.

Remark 7.16 (Large deviations). The large deviations principle for the beta Ginibre gas (2.5) wasestablished in [HP00, PH98], using a method inspired from [BAG08], itself inspired from [Voi93, Voi94].It does not rely on the determinantal structure, and allows to extend Theorem 7.8 to all β > 0.

7.2. More determinantal models. It is well known that the ratio of two independent real standardGaussian random variables follows a Cauchy distribution. The following theorem can be seen as a matrixversion of this phenomenon.

Theorem 7.17 (Forrester – Krishnapur spherical ensemble). Let M1 and M2 be independent copies of theGinibre random matrix defined in (7.1). Then as an exchangeable random vector of Cn, the eigenvaluesof M1M2

−1 have density

(z1, . . . , zn) ∈ Cn 7→ 1

Zn

∏j<k |zj − zk|2

∏nj=1(1 + |zj |2)n+1

.

This corresponds to the beta gas (2.4) with V = 12

n+1n log(1 + |·|2) and β = 2. Moreover its push-forward

on the Riemann sphere using inverse stereographic projection is the uniform law on the sphere.

See [Kri09] and [HKPV09, FK09, For10] for a proof. The set of singular n × n complex matrices is ahyper-surface of zero Lebesgue measure in C

n2

and therefore, almost surely, the Ginibre random matrixM in (7.1) is invertible (its law is absolutely continuous with respect to the Lebesgue measure on Cn2

).From Theorem 1.1, the equilibrium measure of the gas is heavy tailed with density

z ∈ C 7→ 1π(1 + |z|2)2

.

A large deviations principle for the empirical measure associated to the Coulomb gas is proved in [Har12]in relation with the sphere. The convergence of the empirical measure is also considered in [Bor11].The fluctuation at the edge is discussed in [CP14] and studied in [JQ17] by using the idea of Kostlanbehind Theorem 7.13 thanks to the determinantal structure. A beta version of the model is consideredin [CMMOC18] and studied using transportation of measure.

Theorem 7.18 (Życzkowski– Sommers ensemble). Let U = (Uj,k)1≤j,k≤m be a random m × m unitary

matrix following the (Haar) uniform law on this compact group of matrices. Then, for all 1 ≤ n < m, asan exchangeable random vector of Cn, the eigenvalues of the truncation (Uj,k)1≤j,k≤n have density

(z1, . . . , zn) ∈ Cn 7→

∏nj=1(1 − |zj|2)m−n−1

Zn

∏

1≤j<k≤n

|zj − zk|2.

This corresponds to the beta gas (2.4) with V = Vn,m = m−n−1n log 1

1−|z|2 and β = 2.

See [ZS00] for a proof, and [FK09] for the special case m ≥ 2n and a link with the pseudo-sphere andSchur transformation. Following [PR05] and references therein, if limm,n→∞

nm = α ∈ (0, 1) then the

empirical measure converges towards the heavy tailed probability measure with density

z ∈ C 7→ (1 − α)πα(1 − |z|2)2

1|z|≤√α.

In a sense this law interpolates between the uniform law on the unit disc (α → 0 after scaling by√

α)and the uniform law on the unit circle (α → 1). A large deviations principle is obtained in [PR05],concentration inequalities are derived in [MS19], while the fluctuation at the edge is studied in [JQ17].

ASPECTS OF COULOMB GASES 21

Theorem 7.19 (Product of Ginibre random matrices). Let m ≥ 1 and let M1, . . . , Mm be independentand identically distributed copies of the n × n in (7.1). Then, as an exchangeable random vector of Cn,the eigenvalues of the scaled product n− m

2 M1 · · · Mm have density∏n

j=1 wm(√

n|zj |)Zn

∏

j<k

|zj − zk|2

where wk is the Meijer G-function given by the recursive formula

w1(z) = e−|z|2

and wk(z) = 2π

∫ ∞

0

wk−1

(z

r

)e−r2

rdr.

This corresponds to the beta gas (2.4) with V = Vn,m = − 1n log wm(

√n•) and β = 2.

See [AB12] for a proof. Following [GT11, Bor11], its converges to the equilibrium measure with density

z ∈ C 7→ |z| 2m −2

mπ1|z|≤1.

We recover the uniform law on the unit disc when m = 1. The edge fluctuation is considered in [JQ17].

Remark 7.20 (Determinantal gases and random normal matrices). An n × n complex matrix is normalwhen MM∗ = M∗M . The random matrices in theorems 7.1,7.17,7.18,7.19 are not normal. Let us

comment now on models of normal random matrices. Let Nn be the hyper-surface of Cn2

of all n × nnormal matrices. Let V : C → R be C2 and such that V (z) ≥ c log(1 + |z|2) for some constant c > 0.Following [CZ98, EF05], let us consider the probability measure on Nn with density proportional to M 7→e−nTrace(V (M)) with respect to the Hausdorff measure on Nn. This produces random normal matrices, andtheir eigenvalues, seen as an exchangeable random vector, have density given by the gas (2.4) with β = 2.This random (normal) matrix model is referred to as the random normal matrix model. The fluctuationof the empirical measure is studied in [AHM15, AHM11], while the fluctuation at the edge is studied in[CP14, JQ17, GZ18].

The power of a Ginibre matrix has also a nice determinantal structure, see [Dub18a].

8. Comments and open problems

We have skipped several important old and new results on Coulomb gases. The main themes are localversus global, first versus second order, macroscopics versus microscopics, non-universal versus universal.

Universality. The first order global convergence limn→∞ µn = µV , that we call macroscopics, is notuniversal in the sense that the limit µV still depends on V . The second order convergence provided bythe central limit theorem (5.1) is universal in the sense that the limit should not depend on V . Similarly,for a two-dimensional Coulomb gas with radial confining potential V , the limit of the edge depends onV but its fluctuation does not and is universal. Universality emerges often in a second order asymptoticanalysis, as for classical limit theorems of probability theory.

Microscopics. A second order analysis corresponds to the asymptotic analysis of n(µn − νV ) asn → ∞. This can be seen as a microscopic analysis while the convergence µn → µV is a macroscopicanalysis. This corresponds to a second order Taylor formula for the quadratic form EV , in other wordsin a special factorization, leading to a new object called the renormalized energy. This was the subjectof an series of works by Étienne Sandier and Sylvia Serfaty, and by Sylvia Serfaty and other co-authors.See for instance [Ser18b, Ser18a, LS17, Ser15] and references therein. An outcome of this refined analysisis a second order asymptotics for the free energy. More precisely, recall that the Boltzmann – Shannonentropy of the Boltzmann – Gibbs measure Pn in (2.2) is defined by

S(Pn) = −∫

(Rd)n

fn(x1, · · · , xn) log fn(x1, . . . , xn)dx1 · · · dxn

where fn is the density of Pn. Its Helmholtz free energy is given by∫

EndPn − S(Pn)β

= − log Zn

β,

see [Cha15]. Now following [LS17], if µV has density fV with a finite Boltzmann – Shannon entropy

S(µV ) = −∫

Rd

fV log fV dx,

22 DJALIL CHAFAÏ

then we have an asymptotic expansion of the free energy as n → ∞ as

− log Zn

β=

n2

2EV (µV ) − n log n

4+ n(cβ + c′

βS(µV )) + non(1) if d = 2

n2

2EV (µV ) + n(cβ,d,V + c′

βS(µV )) + non(1) if d 6= 2

,

where cβ, c′β , cβ,d,V are constants which can be made explicit.

Edge. The most elementary open question related to Coulomb gases is perhaps the law of fluctuationat the edge, even in the case of rotationally invariant confining potential for arbitrary values of d andβ. The Gumbel fluctuation is known for instance to be universal for a class of two dimensional (d = 2)determinantal (β = 2) Coulomb gases with radial confining potential, see [CP14]. The same question forarbitrary β is open, and the same question for arbitrary dimension d ≥ 3 and β > 0 is also open.

Crystallization. A conjecture related to Coulomb gases is the emergence of rigid structures at lowtemperatures. This is known as crystallization and was proved in special cases, for instance for one-dimensional Coulomb gases. See for instance [Ser15, BL15, Ser18b, Ser18a, PS20] and references therein.

More. Among all the important results on Coulomb gases that we have not yet mentioned, we maycite the approximate transport maps for universality considered in [FG16, BFG15], the rigidity analysisfor hierarchical Coulomb gases considered in [Cha19b], the local density for two-dimensional Coulombgases considered in [BBNY17], the Dobrushin – Lanford – Ruelle equations considered in [DHLM21], theCoulomb gas properties on the sphere considered in [BH19], the local laws and rigidity consideredin [AS21], the quasi-Monte-Carlo method on the sphere considered in [Ber19b], and the Berezinskii –Kosterlitz – Thouless transition [KP17, GS20].

Acknowledgments

We would like to thank David García-Zelada and Kilian Rashel for their helpful remarks.

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